Finding The Range Of The Function Y=√(x+5)

When exploring functions in mathematics, understanding their range is crucial. The range of a function refers to the set of all possible output values (y-values) that the function can produce. In this article, we will delve into the function y=√(x+5) to determine its range. We'll break down the components of the function, discuss the properties of square roots, and use this knowledge to confidently identify the correct range. Understanding the nuances of functions and their ranges is essential for various mathematical applications, making this a fundamental concept for students and enthusiasts alike. Let's embark on this journey to unravel the mysteries of the function y=√(x+5).

H2: Deconstructing the Function y=√(x+5)

To accurately determine the range of the function y=√(x+5), we need to understand its composition. The function is a square root function, which means it involves taking the square root of an expression. The expression inside the square root is (x+5). The square root function has some inherent properties that significantly impact its range. Firstly, the square root of a real number is only defined for non-negative values. This implies that the expression inside the square root, (x+5), must be greater than or equal to zero. This gives us a constraint on the domain of the function, which is the set of all possible input values (x-values). Secondly, the square root of a non-negative number is always non-negative. This property directly affects the possible output values (y-values) of the function. By understanding these two key aspects of the square root function, we can begin to piece together the puzzle of the range of y=√(x+5). The interplay between the input (x) and the resulting output (y) is fundamental to grasping the concept of a function's range. Let's delve deeper into how these properties influence the range of our specific function.

H3: Domain Restrictions and Their Impact

As mentioned earlier, the expression inside the square root, (x+5), must be greater than or equal to zero. This is a fundamental requirement because the square root of a negative number is not defined within the realm of real numbers. Mathematically, we can express this condition as x+5 ≥ 0. Solving this inequality for x, we subtract 5 from both sides, resulting in x ≥ -5. This inequality defines the domain of the function. The domain is the set of all possible x-values that can be input into the function without resulting in an undefined output. In this case, the domain is all real numbers greater than or equal to -5. However, the domain is not the same as the range. The domain restricts the possible inputs, while the range describes the possible outputs. Understanding the domain is a necessary step in determining the range, as it helps us identify the starting point for the output values. Since x must be greater than or equal to -5, this constraint will have a direct impact on the possible values of y. We will explore this connection further as we move towards defining the range.

H3: The Non-Negativity of Square Roots

The square root function has a crucial characteristic: it always produces a non-negative output. This means that the square root of any non-negative number is either zero or a positive number. It is never negative. This inherent property of the square root function is a cornerstone in determining the range of y=√(x+5). Since y is defined as the square root of (x+5), y itself must be non-negative. This immediately tells us that the range of the function cannot include any negative values. Therefore, any options that include negative values in the range can be eliminated. This understanding significantly narrows down the possibilities for the range. We know that y must be greater than or equal to zero, but we need to confirm whether this is the only restriction. To do this, we'll consider how the input values (x) within the domain affect the output values (y). By analyzing the relationship between x and y, we can precisely define the set of all possible y-values, which is the range of the function.

H2: Determining the Range of y=√(x+5) Step-by-Step

Now, let's systematically determine the range of the function y=√(x+5). We've already established that the domain is x ≥ -5 and that the square root function always produces non-negative outputs. This gives us a strong foundation to build upon. To find the range, we need to consider the smallest possible value of y and whether y can take on all values greater than that minimum. When x is at its minimum value within the domain, which is x = -5, we can substitute this value into the function to find the corresponding y-value: y = √(-5+5) = √0 = 0. This tells us that the smallest possible value of y is 0. As x increases from -5, the value of (x+5) also increases, and consequently, the square root of (x+5) also increases. This means that y can take on any non-negative value. There is no upper bound on the value of x, so there is no upper bound on the value of y. Therefore, the range of the function is all real numbers greater than or equal to 0. This understanding is crucial for correctly answering the question and demonstrating a comprehensive understanding of the function's behavior.

H3: Visualizing the Function and its Range

To solidify our understanding, it's helpful to visualize the graph of the function y=√(x+5). The graph starts at the point (-5, 0) and extends upwards and to the right. The shape of the graph is a curve that increases gradually as x increases. The graph never goes below the x-axis, which visually confirms that the y-values are always non-negative. This graphical representation provides a clear picture of the range of the function. We can see that the function covers all y-values greater than or equal to 0. This visual confirmation is a powerful tool for reinforcing the mathematical analysis we've performed. By connecting the algebraic representation of the function with its graphical representation, we gain a more holistic understanding of its behavior and properties. The graph serves as a visual proof of the range we've determined, adding another layer of confidence to our conclusion.

H3: Formal Definition of the Range

Based on our analysis, we can now formally define the range of the function y=√(x+5). The range is the set of all y-values such that y is greater than or equal to 0. We can express this mathematically as y ≥ 0. This concise mathematical statement encapsulates our understanding of the function's output values. It is the definitive answer to the question of the range. This formal definition provides a clear and unambiguous way to communicate the range to others. It also serves as a valuable reference for future mathematical work involving this function. By expressing the range in this way, we demonstrate a mastery of the concept and its mathematical representation. This formal definition is the culmination of our step-by-step analysis and provides a clear and concise answer.

H2: Identifying the Correct Answer and Why Others are Incorrect

Now that we have determined that the range of the function y=√(x+5) is y ≥ 0, we can confidently identify the correct answer from the given options. Option B, y ≥ 0, is the correct answer. Let's analyze why the other options are incorrect.

• Option A, y ≥ -5, is incorrect because the square root function always produces non-negative values. The range cannot include negative values. While the domain is related to -5, this does not translate to the range.
• Option C, y ≥ √5, is incorrect because it suggests that the smallest possible y-value is √5. However, we know that when x = -5, y = 0, which is smaller than √5. This option incorrectly assumes a lower bound on the range.
• Option D, y ≥ 5, is incorrect for a similar reason as option C. It suggests that the smallest possible y-value is 5, which is not true. We have shown that y can take on values between 0 and 5, and the smallest possible value is 0.

By carefully analyzing the function and its properties, we can confidently identify the correct answer and understand why the other options are flawed. This process reinforces our understanding of the range and the importance of considering the function's behavior across its entire domain.

H2: Conclusion: The Range of y=√(x+5) is y ≥ 0

In conclusion, we have thoroughly investigated the function y=√(x+5) and determined its range. By understanding the properties of the square root function, considering the domain restrictions, and systematically analyzing the possible output values, we have definitively shown that the range of the function is y ≥ 0. This means that the function can produce any non-negative value. This understanding is crucial for solving various mathematical problems and gaining a deeper appreciation of functions and their behavior. The process of determining the range involves careful consideration of the function's composition, its domain, and the inherent properties of its operations. By following a step-by-step approach, we can confidently determine the range of even more complex functions. This skill is essential for success in mathematics and related fields. Mastering the concept of range provides a solid foundation for further exploration of mathematical concepts.

This exploration of the range of y=√(x+5) highlights the importance of a methodical approach to problem-solving in mathematics. By breaking down the function into its components, understanding the constraints, and systematically analyzing the possible outputs, we can confidently determine the range. The process of identifying incorrect options further reinforces our understanding of the concept and its nuances. The range is a fundamental property of a function, and mastering its determination is a key step in developing mathematical proficiency. This knowledge empowers us to analyze and understand a wide variety of functions and their applications.